Generation of higher harmonic internal waves by oscillating spheroids

Oscillating bodies in stratified fluids may emit higher harmonics in addition to fundamental waves. In the present experimental study, we consider higher harmonics of an internal wave field generated by a horizontally oscillating spheroid in a linearly stratified fluid for moderate to high oscillation amplitudes, i.e., scaled oscillation amplitude $A/a\ge 0.5$, with $a$ the minor radius of the spheroid. Three different spheroid shapes are tested. The results are discussed in the context of the different theories on the generation of higher harmonics. Higher harmonics are observed at the intersections of fundamental wave beams, and at the critical points of the topography where the topographic slope equals the wave slope. The velocity amplitudes of the fundamental, second, and third harmonic waves grow respectively linearly, quadratically, and with the third power of the scaled oscillation amplitude $A/a$. Though these amplitudes are generally higher when the object's slope is larger, the increase in amplitude above and below the axisymmetric oscillating objects is found to be due to the effect of focusing. In order to discern the relative importance of the harmonics to the fundamental wave, the horizontal structure of the wave amplitude is measured. The results suggest that the $n\text{th}$ harmonic of the internal wave field is associated with a radiation diagram corresponding to a multipole of order $2^n$, with $2n$ directions of propagation.

Figure 1

Fundamental wave (large circle) superposed on the horizontal structure of the (a) dipolar, (b) quadrupolar, and (c) octupolar wave patterns. The annular propagation of the waves and the $2^n$-polar radiation pattern as $\text{cos}\left(n\varphi \right)$, where $\varphi$ is the azimuthal angle, is represented by light gray and the combination of the fundamental wave and higher harmonic is shown in dark gray.

Figure 2

Schematic views of the experimental setup: (a) front view with vertical laser plane; (b) side view with horizontal laser plane. Geometry of oscillating objects: (c) $\mathcal{S}$ sphere; prolate spheroid with (d) vertical axis of revolution $(\mathcal{V}$ spheroid) and (e) horizontal axis of revolution $(\mathcal{H}$ spheroid).

Figure 3

Instantaneous vorticity and velocity vector field of the wave field in the vertical plane of symmetry generated by a $\mathcal{H}$ spheroid (left column) and a $\mathcal{V}$ spheroid (right column): [(a), (b)] total (fundamental and second harmonic), [(c), (d)] first harmonic, and [(e), (f)] second harmonic. Experimental parameters: $N=0.89\mathrm{rad}/\mathrm{s},\mathrm{\Omega }=0.44,A/a=0.5$. Images are presented at the phase $\pm \pi /4$ for the $\mathcal{V}$ and $\mathcal{H}$ spheroids, respectively. Dashed black lines in panels (a) and (b) correspond to extreme positions of spheroids.

Figure 4

Schematic representation of the first and second harmonics of waves generated (according to the theory of Refs. [17, 18, 19]) by (a) the horizontally oriented and (b) vertically oriented prolate spheroids, noted respectively as $\mathcal{H}$ and $\mathcal{V}$ spheroids in the text; green and red dots correspond to critical points ${\varepsilon }_{1,2}=1$, where first and second harmonics are generated. Fundamental (green lines) and second (red dashed lines) harmonic generation from critical points ${\varepsilon }_i$ at the boundary agrees with theory [17]; second harmonic generation from zones of fundamental wave intersection is in agreement with Refs. [18, 19] (blue dashed lines); the harmonic waves that are not observed in the experiment are represented by dashed gray lines. The horizontal dashed gray line determines the vertical position of the fundamental wave beams intersection $Z^{*}$.

Figure 5

Radiation patterns for the first (a) and second harmonic wave (b) of the horizontal longitudinal velocity field at the boundary $\left(Z=-1\right)$ of the oscillating $\mathcal{H}$ spheroid. Experimental parameters $A/a=0.75,N=1.08\mathrm{rad}/\mathrm{s},\text{and}\mathrm{\Omega }=0.4$.

Figure 6

Profiles of second harmonic of horizontal velocity field at $Z=-1$ (red dots), $Z=-1.47$ (green dots), and $Z=-1.91$ (blue dots) for oscillating $\mathcal{H}$ spheroid. Data obtained at $Y=0$ with $A/a=0.75,N=1.08\mathrm{rad}/\mathrm{s},\text{and}\mathrm{\Omega }=0.4$. Here the normalization $X=x/L$ is used, with the length scale $L$ in the direction of oscillation, i.e., $L=b$ in the case of the $\mathcal{H}$ spheroid.

Figure 7

Time-frequency diagram for the waves generated by the $\mathcal{H}$ spheroid, with (a) ${log}_{10}[S_u(t,\omega )/S_u(t,{\omega }_1)]$ averaged over $-2<X<2,-1<Y<1$ in the horizontal plane $Z=-1$ for the horizontal longitudinal velocity $u$, experimental parameters $A/a=0.75$ and $\mathrm{\Omega }=0.4$; and (b) ${log}_{10}[S_w(t,\omega )/S_w(t,{\omega }_1)]$ measured in $-2.5<X<2.5,Y=0$ and $-4<Z<-1$ for the vertical velocity $w$, experimental parameters $A/a=0.5$ and $\mathrm{\Omega }=0.45$.

Figure 8

Distribution of the kinetic energy $E$ defined by Eq. (3) of the first (dots) and second (circles) harmonic for $(X,Y)=(0,0)$ along $Z$. Experimental parameters: $A/a=0.5$ and $\mathrm{\Omega }=0.45$.

Figure 9

Octupolar radiation pattern corresponding to the upper half of Fig. 1 obtained for an oscillating sphere: gray levels visualize (a) the horizontal velocity amplitude of the third harmonic wave $U_a^{\left(3\right)}$ and (b) the vertical velocity amplitude of the third harmonic wave $W_a^{\left(3\right)}$. Experimental parameters: oscillating sphere with $A/a=0.91$ at $Z=-1.66,N=1\mathrm{rad}/\mathrm{s},\mathrm{\Omega }=0.27$.

Figure 10

Profiles of the third harmonic of (a) horizontal $\left(U_a^{\left(3\right)}\right)$ and (b) vertical $\left(W_a^{\left(3\right)}\right)$ velocity field along the $X=x/L$ axis at $Y=0$. Dots, circles, and stars correspond to results for the oscillating $\mathcal{S}$ sphere (measured at $Z=-1.66)$, $\mathcal{V}$ spheroid $\left(Z=-2.4\right)$, and $\mathcal{H}$ spheroid $\left(Z=-1.66\right)$, respectively. The parameter $L=a$ in the cases of $\mathcal{S}$ sphere and $\mathcal{V}$ spheroid and $L=b$ in the case of $\mathcal{H}$ spheroid. Experimental parameters: $A/a=0.65,N=1{\mathrm{s}}^{-1},\text{and}\mathrm{\Omega }=0.27$. (The vertical distance to the boundary of the object is the same in all cases, but $Z$ values vary because of the scaling.)

Figure 11

Variations of the peak magnitudes of the first three harmonics with oscillation amplitude $A/a$ for (a) horizontal $\left(U_a\right)$ and (b) vertical $\left(W_a\right)$ velocity fields. First, second, and third harmonics are represented with red, green, and blue colors, respectively, and compared with theoretical predictions for the sphere (red line) and linear regressions for $U_a^{\left(2\right)}$ $\left(W_a^{\left(2\right)}\right)$ and $U_a^{\left(3\right)}$ $\left(W_a^{\left(3\right)}\right)$ (green and blue lines). Dots, circles, and squares correspond to results for oscillating $\mathcal{S}$ sphere, $\mathcal{V}$ spheroid, and $\mathcal{H}$ spheroid, respectively. $N=1\mathrm{rad}/\mathrm{s},\mathrm{\Omega }=0.27$.